In a multivariate varying-coefficient model, the response vectors Y are regressed on known functions v(X) of some explanatory variables X and the coefficients in an unknown regression matrix ?(Z) depend on another set of explanatory variables Z. We provide statistical tests, called local and global rank tests, which allow one to estimate the rank of an unknown regression coefficient matrix ?(Z) locally at a fixed level of the variable Z or globally as the maximum rank over all levels of Z in a proper, compact subset of the support of Z, respectively. We apply our results to estimate the so-called local and global ranks in a demand system where budget shares are regressed on known functions of total expenditures and the coefficients in a regression matrix depend on prices faced by a consumer.

CEMAT - Center for Computational and Stochastic Mathematics